This result is slightly surprising as the subgroup of \(\displaystyle \mathrm {SL}_{27}\)  generated by \(\displaystyle \mathrm {Spin}_7(\mathbb {C})\)  and \(\displaystyle \mathrm {SL}_3(\mathbb {C})\)  is not \(\displaystyle \mathrm {E}_6\)  (see the above proposition). One also notices that the prehomogeneous space \(\displaystyle (\mathrm {J}_{3}(\mathbb {O}), \mathrm {Aut}^{0}(\mathfrak {S})) \times \mathbb {C}^*\)  does not appear in the list of Kimura-Sato [1]}. This is not surprising as it is unlikely to be an irreducible prehomogeneous space. It thus provides an interesting example of a non-irreducible prehomogeneous vector space which does not split as a sum of lower dimensional prehomoegeneous vector spaces. Theorem REF  also shows that the geometry of the twisted octonionic eigenvalue problem differes eminently from that of the untwisted eigenvalue problem : the former is much more <<symmetric>> than the latter. Indeed, we noticed in section 2.2 that \(\displaystyle  \mathrm {Aut}^{0}(\mathrm {S_{ODM}}) \times \mathbb {C}^* = \mathrm {SO}_7 \times \mathrm {SL}_3 \times \mathbb {C}^*\)  does not act prehomogeneously on \(\displaystyle \mathrm {J}_{3}(\mathbb {O})\) .
